Free energy and extension of a semiflexible polymer in cylindrical confining geometries

We consider a long, semiflexible polymer, with persistence length $P$ and contour length $L$, fluctuating in a narrow cylindrical channel of diameter $D$. In the regime $D\ll P\ll L$ the free energy of confinement $\mathrm{\Delta }F$ and the length of the channel $R_{\parallel }$ occupied by the polymer are given by Odijk’s relations $\mathrm{\Delta }F\mathrm{/}{R}_{\parallel }=A_{\circ }{k}_{B}T{P}^{-1\mathrm{/}3}{D}^{-2\mathrm{/}3}$ and $R_{\parallel }=L[1-{\alpha }_{\circ }(D\mathrm{/}P{)}^{2\mathrm{/}3}]$, where $A_{\circ }$ and ${\alpha }_{\circ }$ are dimensionless amplitudes. Using a simulation algorithm inspired by PERM (Pruned Enriched Rosenbluth Method), which yields results for very long polymers, we determine $A_{\circ }$ and ${\alpha }_{\circ }$ and the analogous amplitudes for a channel with a rectangular cross section. For a semiflexible polymer confined to the surface of a cylinder, the corresponding amplitudes are derived with an exact analytic approach. The results are relevant for interpreting experiments on biopolymers in microchannels or microfluidic devices.